Divine Proportions: Rational Trigonometry to Universal Geometry

Introduction
I Preliminaries
1 Overview
1.1 Introducing quadrance and spread
1.2 Laws of rational trigonometry
1.3 Why classical trigonometry is hard
1.4 Why rational trigonometry is easier
1.5 Comparison example
1.6 Ancient Greek triumphs and difficulties
1.7 Modern ambiguities
2 Background
2.1 Fields
2.2 Proportions
2.3 Identities and determinants
2.4 Linear equations
2.5 Polynomial functions and zeroes
2.6 Quadratic equations
3 Cartesian coordinate geometry
3.1 Points and lines
3.2 Collinear points and concurrent lines
3.3 Parallel and perpendicular lines
3.4 Parallels and altitudes
3.5 Sides, vertices and triangles
3.6 Quadrilaterals
3.7 Affine combinations
3.8 Perpendicular bisectors
4 Reflections
4.1 Affine transformations
4.2 Lineations and reflection sequences
II Rational trigonometry
5 Quadrance
5.1 Quadrances of triangles and quadrilaterals
5.2 Triple quad formula
5.3 Pythagoras’ theorem
5.4 Quadrance to a line
5.5 Quadrea
5.6 Archimedes’formula
5.7 Quadruple quad formula
6 Spread
6.1 Spreads of triangles and quadrilaterals
6.2 Cross
6.3 Twist
6.4 Ratio theorems
6.5 Complementary spreads
6.6 Spread law
6.7 Cross law
6.8 Spreads in coordinates
6.9 Vertex bisectors
7 Triple spread formula
7.1 Triple spread formula
7.2 Triple cross formula
7.3 Triple twist formula
7.4 Equal spreads
7.5 Spread reflection theorem
7.6 Examples using different fields
7.7 Quadruple spread formula
8 Spread polynomials
8.1 Combining equal spreads
8.2 Spread polynomials
8.3 Special cases
8.4 Explicit formulas
8.5 Orthogonality . . .
8.6 Composition of spread polynomials
8.7 Cross polynomials
9 Oriented triangles and turns
9.1 Oriented sides, vertices and triangles
9.2 Turns of oriented vertices
9.3 Signed areas
III Universal Geometry
10 Triangles
10.1 Isosceles triangles
10.2 Equilateral triangles
10.3 Right triangles
10.4 Congruent and similar triangles
10.5 Solving triangles
11 Laws of proportion
11.1 Triangle proportions
11.2 Quadrilateral proportions
11.3 Two struts theorem
11.4 Stewart’s theorem
11.5 Median quadrance and spread
11.6 Menelaus’and Ceva’stheorems
12 Centers of triangles
12.1 Perpendicular bisectors and circumcenter
12.2 Formulas for the circumcenter
12.3 Altitudes and orthocenter
12.4 Formulas for the orthocenter
12.5 Incenters
13 Isometries
13.1 Translations, rotations, reflections
13.2 Classifying isometries
14 Regular stars and polygons
14.1 Regular stars
14.2 Order three stars
14.3 Order five stars
14.4 Order seven stars
14.5 Regular polygons
15 Conics
15.1 Centers of conics
15.2 Circles and ribbons
15.3 Parabolas
15.4 Quadrolas
15.5 Grammolas
15.6 Intersections with lines
16 Geometry of circles
16.1 Diameters and chords
16.2 Spreads in a circle
16.3 Parametrizing circles
17 Quadrilaterals
17.1 Cyclic quadrilaterals
17.2 Circumquadrance formula
17.3 Cyclic quadrilateral quadrea
17.4 Ptolemy’s theorem
17.5 Four point relation
18 Euler line and nine point circle
18.1 Euler line
18.2 Nine point circle
19 Tangent lines and tangent conics
19.1 Translates and Taylor conics
19.2 Tangent lines
19.3 Higher order curves and tangents
19.4 Folium of Descartes
19.5 Lemniscate of Bernoulli
IV Applications
20 Triangle spread rules
20.1 Spread ruler
20.2 Line segments,rays and sectors
20.3 Acute and obtuse sectors
20.4 Acute and obtuse triangles
20.5 Triangle spread rules
21 Two dimensional problems
21.1 Harmonic relation
21.2 Overlapping triangles
21.3 Eyeball theorem
21.4 Quadrilateral problem
22 Three dimensional problems
22.1 Planes
22.2 Boxes
22.3 Pyramids
22.4 Wedges
22.5 Three dimensional Pythagoras’ theorem
22.6 Pagoda and seven-fold symmetry
23 Physics applications
23.1 Projectile motion
23.2 Algebraic dynamics
23.3 Snell’s law
23.4 Lorentzian addition of velocities
24 Surveying
24.1 Height of object with vertical face
24.2 Height of object with inaccessible base
24.3 Height of a raised object
24.4 Regiomontanus’problem
24.5 Height from three spreads
24.6 Vertical and horizontal spreads
24.7 Spreads over a right triangle
24.8 Spherical analogue of Pythagoras’ theorem .
25 Resection and Hansen’s problem
25.1 Snellius-Pothenot problem
25.2 Hansen’s problem
26 Platonic solids
26.1 Tetrahedron
26.2 Cube
26.3 Octahedron
26.4 Icosahedron
26.5 Dodecahedron
27 Rational spherical coordinates
27.1 Polar spread and quadrance
27.2 Evaluating pi^2/16
27.3 Beta function
27.4 Rational spherical coordinates
27.5 Surface measure on a sphere
27.6 Four dimensional rational spherical coordinates
27.7 Conclusion
A Rational polar equations of curves
B Ellipson
Theorems with pages and Important Functions